The Astrophysical Journal, 780:167 (10pp), 2014 January 10 doi:10.1088/0004-637X/780/2/167C© 2014. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

PROPERTIES OF AN ECLIPSING DOUBLE WHITE DWARF BINARY NLTT 11748

David L. Kaplan1, Thomas R. Marsh2, Arielle N. Walker1, Lars Bildsten3,4,Madelon C. P. Bours2, Elme Breedt2, Chris M. Copperwheat5, Vik S. Dhillon6, Steve B. Howell7,

Stuart P. Littlefair6, Avi Shporer8, and Justin D. R. Steinfadt41 Physics Department, University of Wisconsin–Milwaukee, Milwaukee, WI 53211, USA; kaplan@uwm.edu.

2 Department of Physics, University of Warwick, Coventry CV4 7AL, UK3 Kavli Institute for Theoretical Physics and Department of Physics, Kohn Hall, University of California, Santa Barbara, CA 93106, USA

4 Department of Physics, Broida Hall, University of California, Santa Barbara, CA 93106, USA5 Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK

6 Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK7 NASA Ames Research Center, Moffett Field, CA 94035, USA

8 California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USAReceived 2013 July 19; accepted 2013 November 21; published 2013 December 20

ABSTRACT

We present high-quality ULTRACAM photometry of the eclipsing detached double white dwarf binary NLTT11748. This system consists of a carbon/oxygen white dwarf and an extremely low mass (<0.2 M�) helium-corewhite dwarf in a 5.6 hr orbit. To date, such extremely low-mass white dwarfs, which can have thin, stably burningouter layers, have been modeled via poorly constrained atmosphere and cooling calculations where uncertaintiesin the detailed structure can strongly influence the eventual fates of these systems when mass transfer begins.With precise (individual precision ≈1%), high-cadence (≈2 s), multicolor photometry of multiple primary andsecondary eclipses spanning >1.5 yr, we constrain the masses and radii of both objects in the NLTT 11748 systemto a statistical uncertainty of a few percent. However, we find that overall uncertainty in the thickness of the envelopeof the secondary carbon/oxygen white dwarf leads to a larger (≈13%) systematic uncertainty in the primary HeWD’s mass. Over the full range of possible envelope thicknesses, we find that our primary mass (0.136–0.162 M�)and surface gravity (log(g) = 6.32–6.38; radii are 0.0423–0.0433 R�) constraints do not agree with previousspectroscopic determinations. We use precise eclipse timing to detect the Rømer delay at 7σ significance, providingan additional weak constraint on the masses and limiting the eccentricity to e cos ω = (−4 ± 5)×10−5. Finally, weuse multicolor data to constrain the secondary’s effective temperature (7600±120 K) and cooling age (1.6–1.7 Gyr).

Key words: binaries: eclipsing – stars: individual (NLTT 11748) – techniques: photometric – white dwarfs

Online-only material: color figures

1. INTRODUCTION

Among the more interesting products of binary evolution arecompact binaries (periods less than 1 day) containing helium-core white dwarfs (WDs). These WDs are created from low-mass (<2.0 M�) stars when stellar evolution is truncated by abinary companion before the He core reaches the ≈0.48 M�needed for the helium core flash. Such WDs were first identifiedboth as companions to millisecond pulsars (Lorimer et al. 1995;van Kerkwijk et al. 2005; Bassa et al. 2006) and other WDs (e.g.,Bergeron et al. 1992; Marsh et al. 1995), with large numbers ofdouble WD binaries discovered in recent years. In particular, theExtremely Low Mass (ELM) survey (Kilic et al. 2012; Brownet al. 2013, and references therein) has discovered tens of newHe-core WDs in the last few years, focusing on the objects withmasses <0.2 M�.

The compact binaries containing these WDs will inspiral dueto emission of gravitational radiation in less than a Hubble time;the most compact of them will merge in <1 Myr (Brown et al.2011). When mass transfer begins, detailed evolutionary andmass transfer calculations (Marsh et al. 2004; D’Antona et al.2006; Kaplan et al. 2012) will determine whether the objectsremain separate (typically resulting in an AM CVn binary) ormerge (as a R CrB star or possibly a Type Ia supernova; Iben& Tutukov 1984; Webbink 1984). Essential to determining thefates of these systems (and hence making predictions for low-frequency gravitational radiation and other end products) is an

accurate knowledge of their present properties: their massesdetermine the in-spiral time and their radii and degrees ofdegeneracy help determine the stability of mass transfer (Deloyeet al. 2005; D’Antona et al. 2006; Kaplan et al. 2012). This isparticularly interesting for the ELM WDs, as they are predictedto possess stably burning H envelopes (with ∼10−3–10−2 M�of hydrogen) that keep them bright for Gyr (Serenelli et al.2002; Panei et al. 2007) and increase their radii compared with“cold,” fully degenerate WDs by a factor of two or more. Thisburning slows the cooling behavior of these objects (it may notbe monotonic for all objects) and improving our understandingof ELM WD cooling would aid in evolutionary models formillisecond pulsars and later stages of mass transfer (e.g., Tauriset al. 2012; Antoniadis et al. 2012; Kaplan et al. 2013).

Few ELM WDs have had mass and radius measurements ofany precision. As most systems are single-line spectroscopicbinaries (Kaplan et al. 2012), precise masses are difficult toobtain (although some pulsar systems are better; Bassa et al.2006; Antoniadis et al. 2012). Radii are even harder, typicallyrelying on poorly calibrated surface gravity measurements andcooling models (as in Kilic et al. 2012). The eclipsing doubleWD binary NLTT 11748 (Steinfadt et al. 2010) allowed for thefirst geometric measurement of the radius of an ELM WD in thefield (cf. PSR J1911−5958A in the globular cluster NGC 6752;Bassa et al. 2006), finding R ≈ 0.04 R� for the ≈0.15 M� HeWD, with new eclipsing systems (Parsons et al. 2011; Brownet al. 2011; Vennes et al. 2011) helping even more. However,

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The Astrophysical Journal, 780:167 (10pp), 2014 January 10 Kaplan et al.

Table 1Log of ULTRACAM Observations and Eclipse Times

Date Eclipse Time Telescope Eclipse Filtersa Exposuresa Num. Starsa Precisionsa

(MBJD) (s) (%)

2010 Nov 12 . . . 55512.182179(17) NTT secondary u′g′i′ 7.69, 2.55, 2.55 4, 5, 3 2.9, 1.0, 1.32010 Nov 15 . . . 55515.120443(21) NTT primary u′g′i′ 7.69, 2.55, 2.55 4, 5, 4 6.4, 1.8, 2.22010 Nov 15 . . . 55515.237910(23) NTT secondary u′g′i′ 7.69, 2.55, 2.55 4, 5, 3 3.6, 1.2, 1.72010 Nov 25 . . . 55525.228090(18) NTT primary u′g′i′ 7.69, 2.55, 2.55 4, 5, 3 4.2, 1.5, 1.62010 Nov 26 . . . 55526.168300(13) NTT primary u′g′r ′ 5.89, 1.95, 1.95 4, 5, 3 3.5, 1.3, 1.62010 Nov 26 . . . 55526.285767(29) NTT secondary u′g′r ′ 5.89, 1.95, 1.95 4, 5, 2 6.6, 1.9, 2.52010 Nov 27 . . . 55527.108548(12) NTT primary u′g′i′ 5.89, 1.95, 1.95 4, 5, 4 3.4, 1.1, 1.52010 Nov 27 . . . 55527.226026(20) NTT secondary u′g′i′ 5.46, 1.35, 1.35 4, 5, 4 3.8, 1.5, 1.82010 Nov 28 . . . 55528.166253(16) NTT secondary u′g′r ′ 5.89, 1.95, 1.95 4, 5, 5 3.3, 1.1, 1.32010 Nov 29 . . . 55529.106526(19) NTT secondary u′g′i′ 5.46, 1.35, 1.35 4, 4, 4 4.1, 1.4, 2.02010 Dec 2 . . . 55532.162334(16) NTT secondary u′g′i′ 7.69, 2.55, 2.55 4, 4, 4 2.6, 0.9, 1.32010 Dec 10 . . . 55540.154399(16) NTT secondary u′g′i′ 7.69, 2.55, 2.55 4, 4, 4 2.7, 0.9, 1.32010 Dec 15 . . . 55545.208237(16) NTT primary u′g′i′ 7.48, 2.48, 2.48 4, 4, 4 3.7, 1.3, 1.72010 Dec 16 . . . 55546.148475(18) NTT primary u′g′i′ 7.48, 2.48, 2.48 4, 4, 4 4.7, 1.6, 2.02010 Dec 17 . . . 55547.088715(23) NTT primary u′g′i′ 7.48, 2.48, 2.48 4, 4, 4 5.1, 1.8, 2.12010 Dec 18 . . . 55548.146459(36) NTT secondary u′g′i′ 7.48, 2.48, 2.48 4, 4, 4 7.0, 2.6, 3.42012 Jan 17 . . . 55943.870809(09) WHT primary u′g′r ′ 2.48, 2.48, 2.48 4, 5, 3 4.1, 0.7, 0.82012 Jan 17 . . . 55943.988273(13) WHT secondary u′g′r ′ 4.99, 2.48, 2.48 4, 5, 4 2.9, 0.8, 0.92012 Jan 18 . . . 55944.928546(22) WHT secondary u′g′r ′ 4.99, 2.48, 2.48 4, 5, 3 4.1, 1.2, 1.52012 Jan 19 . . . 55945.046118(28) WHT primary u′g′r ′ 4.98, 2.48, 2.48 4, 5, 3 5.3, 1.7, 2.02012 Jan 19 . . . 55945.868766(42) WHT secondary u′g′ 5.99, 2.98, · · · 4, 5, · · · 3.7, 1.3, · · ·2012 Jan 21 . . . 55947.866825(09) WHT primary u′g′r ′ 3.46, 1.72, 1.72 4, 5, 2 2.9, 0.9, 1.02012 Jan 22 . . . 55948.924551(12) WHT secondary u′g′r ′ 3.99, 1.98, 1.98 4, 5, 4 2.9, 0.8, 0.92012 Jan 23 . . . 55949.042137(10) WHT primary u′g′r ′ 3.99, 1.98, 1.98 4, 5, 4 4.0, 0.9, 1.12012 Sep 1 . . . 56171.174286(11) WHT primary u′g′r ′ 5.56, 1.84, 1.84 4, 4, 4 3.8, 1.6, 1.12012 Sep 4 . . . 56174.230072(11) WHT primary u′g′r ′ 6.48, 2.14, 2.14 5, 4, 4 2.5, 1.0, 1.02012 Sep 10 . . . 56180.106572(12) WHT primary u′g′r ′ 5.00, 2.48, 2.48 4, 4, 4 2.8, 2.1, 0.9

Note. a We give the three filters used along with the corresponding exposure times, number of reference stars used, and typical fractional precisions on a singlemeasurement of NLTT 11748.

for NLTT 11748 the original eclipse constraints from Steinfadtet al. (2010) were limited in their precision. As the system is asingle-line binary, individual masses were not known. Furtheruncertainties came from limited photometric precision and alow observational cadence, along with ignorance of proper limbdarkening for WDs of this surface gravity and temperature.

Here, we present new data and a new analysis of eclipsephotometry for NLTT 11748 that rectifies almost all of theprevious limitations and gives precise masses and radii thatare largely model independent (at least concerning models of theELM WDs themselves), allowing for powerful new constraintson the evolution and structure of ELM WDs. NLTT 11748 wasidentified by Kawka & Vennes (2009) as a candidate ELMWD binary, containing a helium-core WD with mass ≈0.15 M�presumably orbiting with a more typical 0.6 M� carbon/oxygen(CO) WD (note that the photometric primary is the lower-mass object, owing to the inverted WD mass-radius relation).While searching for pulsations, Steinfadt et al. (2010) foundperiodic modulation in the light curve of NLTT 11748 whichthey determined was due to primary (6%) and secondary (3%)eclipses in a 5.6 hr orbit, as confirmed by radial velocitymeasurements (also see Kawka et al. 2010). The primary low-mass WD has a low surface gravity (log(g) = 6.18 ± 0.15from Kawka et al. 2010, log(g) = 6.54 ± 0.05 from Kilic et al.2010) and an effective temperature Teff = 8580 ± 50 K (Kawkaet al. 2010 or 8690 ± 140 K from Kilic et al. 2010). Constraintsin this region of (log(g), Teff) space are particularly valuable,as the behavior of systems in this region is complex with awide range of predicted ages consistent changing over smallranges of mass, especially since it is near the transition from

systems that show CNO flashes and those that do not (Althauset al. 2013).

Our new data consist of high-cadence (2.5 s compared with30 s previously), high-precision photometry in multiple simul-taneous filters, which we combine with improved modelingand knowledge of limb-darkening coefficients (Gianninas et al.2013). We outline the new observations in Section 2. The ma-jority of the new analysis is described in Section 3, with theresults in Section 3.2. Finally, we make some additional physi-cal inferences and discuss our results in Section 4.

2. OBSERVATIONS AND REDUCTION

2.1. ULTRACAM Observations

We observed NLTT 11748 with ULTRACAM (Dhillon et al.2007) over 27 eclipses during 2010 and 2012, as summarized inTable 1. ULTRACAM provides simultaneous fast photometrythrough 3 filters with negligible dead time. During 2010, it wasmounted on the 3.5 m New Technology Telescope (NTT) at LaSilla Observatory, Chile. We used the u′ and g′ filters, alongwith either r ′ or i ′. The integration times were chosen based onthe conditions, but were typically 1–2 s for the redder filters and5–8 s for the u′ filter. During 2012, ULTRACAM was mountedon the 4.2 m William Herschel Telescope at the Observatoriodel Roque de los Muchachos on the island of La Palma. Here,we used only the u′g′r ′ filters, although for one observation wediscarded the r ′ data as they were corrupted. Exposure timeswere 2–3 s for the redder filters and 3–5 s for the u′ filter, takingadvantage of better conditions, a larger mirror, and a lower

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The Astrophysical Journal, 780:167 (10pp), 2014 January 10 Kaplan et al.

airmass toward this northern target. The total observing time foreach eclipse was typically less than 40 minutes.

The data were reduced using custom software. We firstdetermined bias and flatfield images appropriate for everyobservation. Then, we measured aperture photometry for NLTT11748 and up to six reference sources that were typicallysomewhat brighter than NLTT 11748 itself. The aperture wassized according to the mean seeing for the observation, but washeld fixed for a single observation and a single filter. Finally, aweighted mean of the reference star magnitudes (some of whichwere removed because of saturation and some of which wereremoved because of low signal-to-noise, especially in the u′band) was subtracted from the measurements for NLTT 11748.These detrended data were the final relative photometry that weused in all subsequent analysis. Given the short duration of theeclipse (≈3 minutes), any remaining variations in the relativephotometry due to transparency or airmass changes could beignored; we did not attempt to model the out-of-eclipse data(cf. Shporer et al. 2010).

2.2. Near-infrared Observations

In addition to the ULTRACAM observations, we observeda few eclipses using the Gemini Near-Infrared Imager (NIRI;Hodapp et al. 2003) on the 8 m Gemini-North telescope un-der program GN-2010B-Q-54. Given the low signal-to-noiseand relatively long cadence, the primary eclipse data werenot particularly useful in constraining the properties of thesystem. Instead, we concentrate on the secondary eclipse ob-servations, where the additional wavelength coverage is help-ful in constraining the effective temperature of the secondary(Section 4.2). The data were taken on 2010 November 21 and2010 December 15 using the J-band filter. The total duration ofthe observations were 37 and 31 minutes around a secondaryeclipse, as predicted from our initial ephemeris. Successive ex-posures happened roughly every 25 s, of which 20 s were actu-ally accumulating data, so our overhead was about 20%.

To reduce the data, we used the nprepare task in IRAF, whichadds various meta-data to the FITS files. We then corrected thedata for nonlinearities9 and applied flatfields computed usingthe niflat task, which compared dome-flat exposures takenwith the lamp on and off to obtain the true flatfield. We used ourown routines to perform point spread function photometry witha Moffat (1969) function. This function was held constant foreach observation and was fit to bright reference stars. Finally,we subtracted the mean of two bright reference stars to de-trendthe data.

3. ECLIPSE FITTING

To start, we determined rough eclipse times and shapes byfitting a simple model to the data, using a square eclipse forthe secondary eclipses and a linear limb-darkening law for theprimary eclipses. These results were only used as a starting pointfor the later analysis, but the times were correct to ±10 s. Wethen fit the photometry data, as summarized below. The fittingused only the ULTRACAM data; the NIRI data were added laterto constrain the secondary temperature.

Our main eclipse fitting used a Markov-Chain Monte Carlo(MCMC) fitter, based on a Python implementation10 (Foreman-Mackey et al. 2013) of the affine-invariant ensemble sampler

9 Using the nirlin.py script fromhttp://staff.gemini.edu/∼astephens/niri/nirlin/nirlin.py.10 See http://dan.iel.fm/emcee/.

(Goodman & Weare 2010). We parameterized the light curveaccording to

1. Mass and radius of the primary (low-mass) WD, M1 and R1

2. Orbital inclination i3. Radial velocity amplitude of the primary K1

4. Mean period PB, reference time t0, and time delay Δt

5. Temperatures of the primary T1 and secondary T2

for nine total parameters. These were further constrained bypriors based on spectroscopy, with K1 = 273.4 ± 0.5 km s−1

and T1 = 8690±140 K (Steinfadt et al. 2010; Kilic et al. 2010).We assumed a strictly periodic ephemeris (with no spin down;see Section 4.1) that includes a possible time delay between theprimary and secondary eclipses (Kaplan 2010).

Our limb-darkening law used four-parameter (Claret 2000)limb-darkening coefficients, as determined by Gianninas et al.(2013) for a range of gravities and effective temperatures. Weinterpolated the limb-darkening parameters for the primary’stemperature T1, although we used several fixed values of log(g)(6.25, 6.50, and 6.75) instead of the value for each fit. This isdone both to avoid numerical difficulties in two-dimensional(2D) interpolation over a coarse grid and to avoid biasing thefitted log(g) by anything other than the light-curve shape (unlikethe temperature, the different spectroscopic determinations ofthe surface gravity are significantly discrepant). We found thatvariations in log(g) used for the limb-darkening parameterschanged the fit results by <1σ .

With values for M1, i, PB, and K1, the mass of the secondary(high-mass) WD, M2, is then determined (and so is the massratio q ≡ M1/M2, as well as K2 = qK1). The final parameteris the radius of the secondary WD R2. However, as this isa more or less normal CO WD that is not tidally distorted((M1/M2)(R2/a)3 ≈ 10−7), we used a mass-radius relationappropriate for WDs in this mass range (Fontaine et al. 2001;Bergeron et al. 2001),11 interpolating linearly. The complicationthat this introduced is that WDs with finite temperatures do haveradii slightly larger than the nominal zero-temperature modeland that this excess depends on the thickness of their hydrogenenvelopes. For the mass and temperature range considered here,this excess is typically r2 ≡ R2/R2(T2 = 0) = 1.02–1.06. Inwhat follows, we treat r2 as a free parameter and give our resultsin terms of r2, with a detailed discussion of the influence of r2on the other parameters in Section 3.1.

Overall, we had 22,574 photometry measurements within±400 s of the eclipses (shown in Figure 1), which we correctedto the solar system barycenter using a custom extension tothe TEMPO212 pulsar timing package (Hobbs et al. 2006). Theeclipses themselves were modeled with the routines of Agol(2002; also see Mandel & Agol 2002), which accounts forintrabinary lensing (Maeder 1973; Marsh 2001).

We started 200 MCMC “walkers,” where each walker ex-ecutes an independent path through the parameter space. Thewalkers were initialized from normally distributed random vari-ables, with each variable taken from the nominal values de-termined previously (Steinfadt et al. 2010; Kilic et al. 2010)with generous uncertainties. In the end, we increased the un-certainties on the initial conditions and it did not change theresulting parameter distributions. Each walker was allowed 500iterations to “burn in,” after which its memory of the sampledparameter space was deleted. Finally, each walker iterated for

11 See http://www.astro.umontreal.ca/∼bergeron/CoolingModels/.12 See http://www.atnf.csiro.au/research/pulsar/tempo2/.

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The Astrophysical Journal, 780:167 (10pp), 2014 January 10 Kaplan et al.

−0.010 −0.005 0.000 0.005 0.010Orbital Phase (cycles)

0.8

0.9

1.0

1.1

1.2

1.3R

elat

ive

Flu

x(p

lus

cons

tant

)

1 min

u’

g’

r’

i’

Primary

0.495 0.500 0.505 0.510Orbital Phase (cycles)

0.8

0.9

1.0

1.1

1.2

1.3

Rel

ativ

eF

lux

(plu

sco

nsta

nt)

u’

g’

r’

i’

Secondary

Figure 1. Normalized primary (left) and secondary (right) eclipses of NLTT 11748, as measured with ULTRACAM. The raw data are the points, while binned dataare the circles with error bars and the best-fit models are the solid lines. The different filters are labeled. A 1 minute interval is indicated by the scale bar at the lowerleft. Data from 2010 and 2012 have been combined.

(A color version of this figure is available in the online journal.)

a further 5000 cycles, giving 200 × 5000 = 1, 000, 000 sam-ples. However, not all of these are independent: we measuredan auto-correlation length of about 100 samples from the re-sulting distributions, so we thinned the parameters by takingevery 91 samples (we wanted a number near our measured auto-correlation but that was not commensurate with the number ofwalkers).

3.1. The Influence of the Secondary’s EnvelopeThickness on the Measurement

As discussed above, our one significant assumption (whichwas also made in Steinfadt et al. 2010) is that the secondarystar follows the mass-radius relation for a CO WD. This seemsreasonable, given inferences from observations (Kawka et al.2010; Kilic et al. 2010) and from evolutionary theory. However,with the high precision of the current dataset, we must exam-ine the choice of mass-radius relation closely. In particular, thezero-temperature model used in Steinfadt et al. (2010) is nolonger sufficient. For effective temperatures near 7500 K andmasses near 0.7 M�, finite-temperature models are larger thanthe zero-temperature models by roughly 2% (thin envelopes,taken to be 10−10 of the star’s mass) to 6% (thick envelopes,taken to be 10−4 of the star’s mass) and moreover the excessdepends on mass (Fontaine et al. 2001; Bergeron et al. 2001).This excess is similar to what we observe in a limited seriesof models computed using Modules for Experiments inStellar Astrophysics (Paxton et al. 2011, 2013). The en-velope thickness can be constrained directly through astroseis-mology of pulsating WDs (ZZ Ceti stars), with most sourceshaving thick envelope (fractional masses of 10−6 or above), butextending down such that roughly 10% of the sources have thinenvelopes (fractional masses of 10−7 or below); there is a peakat thicker envelopes, but there is a broad distribution (Romero

et al. 2012; consistent with the findings of Tremblay & Bergeron2008).

We use the parameter r2 to explore the envelope thickness.Values near 1.02 correspond to thin envelopes (with some slightmass dependence), while values near 1.06 correspond to thickenvelopes. Changing r2 over the range of values discussed aboveleads to changes in the best-fit physical parameters M1, M2,R1, and R2. In particular, M1 was surprisingly sensitive to r2.Determining a “correct” value for r2 is beyond the scope of thiswork, but we can understand how the physical parameters scalewith r2 in a reasonably simple manner. Since we know the periodaccurately and can also say that sin i ≈ 1 (deeply eclipsing), weknow that M1 + M2 ∝ a3 (where a is the semi-major axis) fromKepler’s third law. We also know K1, which is the orbital speedof the primary:

K1 ≈ 2πa

PB

M2

M1 + M2, (1)

which allows us to constrain aM2 ∝ M1 +M2. Combining thesegives M2 ∝ a2. We can further parameterize the mass-radiusrelation of the secondary:

R2 ∝ r2Mβ

2 . (2)

The duration of the eclipse fixes R1/a, while the duration ofingress/egress fixes R2/a (e.g., Winn 2011), so we can also saya ∝ R2 or a ∝ r2M

β

2 . Combining this with M2 ∝ a2 givesM2 ∝ r

2/(1−2β)2 . Near M2 ≈ 0.7 M�, the mass–radius relation

has a slope β ≈ −0.78 (compare with the traditional β = −1/3for lower masses), so M2 ∝ r0.78

2 . Since a ∝ M1/22 and both R1

and R2 are ∝ a, R1 ∝ r1/(1−2β)2 ∝ r0.39

2 and the same for R2 (at afixed M2, R2 ∝ r2, as given in Equation(2); however, the resulthere is for the best-fit value of R2, which can result in changesof the other parameters including M2).

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The Astrophysical Journal, 780:167 (10pp), 2014 January 10 Kaplan et al.

0.10 0.15M1 (M )

700074007800

T 2(K

)

02468

Δt(s

)

−2−1

01

ΔPB

(μs)

0.0020.0050.0080.011

cos(

i)

0.670.710.75

M2

(M) 0.040

0.0420.0440.046

R1

(R)

M1 = 0.136 ± 0.007 M

0.041 0.046R1 (R )

R1 = 0.0423 ± 0.0004 R

0.680.72M2 (M )

M2 = 0.707 ± 0.008 M

0.002 0.012cos(i)

cos(i) = 0.0058 ± 0.0021

−1 0 1ΔPB (μs)

ΔPB = -0.0 ± 0.3 μs

0 4 8Δt (s)

Δt = 4.2 ± 0.6 s

7200 7800T2 (K)

T2 = 7597 ± 120 K

Figure 2. Joint confidence contours on the parameters from the fit of NLTT 11748, assuming r2 = 1.00. We show 68%, 95%, and 99.7% contours on 2D distributionsthat have been marginalized from the 8D original distribution (we do not plot distributions for the reference time t0, as it is of little physical interest). ΔPB is the offsetof PB with respect to its mean (Table 2). We also show the 1D distributions for each parameter. In all of the plots, the black dashed lines show the means and the blackdotted lines show the ±1σ limits. Finally, in the plot of M2 vs. M1, we show solid lines corresponding to the mass ratios q = 0.16, 0.18, and 0.20 (which can map toconstraints from Δt), while in the plot of R1 vs. M1, we show solid lines corresponding to log(g) = 6.2, 6.3, and 6.4.

(A color version of this figure is available in the online journal.)

To understand how M1 changes with r2, we use the Keplerianmass function, which fixes M3

2 ∝ (M1 + M2)2. If we take thelogarithmic derivative of this, we find:

d log M1

M1= α

d log M2

M2, (3)

with α = 3/2 + M2/2M1. Since our mass ratio q is roughly0.2 (Table 2), we find α ≈ 4.0. From this, we find thatM1 ∝ r

2α/(1−2β)2 ∝ r3.13

2 , which is much steeper than the otherdependencies. These relations are borne out by MCMC results(Table 2, Figures 2 and 3).

3.2. Results

The model fit the data well, with a minimum χ2 of 22978.8for 22,566 degrees of freedom (dof).13 We show 1D and 2Dmarginalized confidence contours in Figure 2 and the best-fitlight curves in Figure 1. The results are given in Table 2. Alinear ephemeris gives a satisfactory fit to the data, although

13 In all of our fittings, we make use of the χ2 statistic. This assumes thatindividual data points are independent of each other; on the other hand,correlated errors can become significant for very precise photometry and canalter the nature of parameter and uncertainty estimation (see Carter & Winn2009). To test this, we examined the out-of-eclipse data for any correlationbetween subsequent data points. We found autocorrelation lengths of 0–2samples, with a mean of 0.85. This was very similar to the distribution ofautocorrelation lengths estimated from sets of 100 uncorrelated randomnumbers drawn from N (1, 0.003) (similar in length and properties to our data),so we conclude that deviations from an autocorrelation length of 0 areconsistent with the finite sample sizes that we used and that the data areconsistent with being independent.

we find a significantly non-zero value for Δt , which we discussbelow.

For NIRI, the quality of the data is modest, with typicaluncertainties of ±0.03 mag and a cadence of 25 s. Given thequality of the ULTRACAM results, fitting the NIRI data withall parameters free would not add to the results. Instead, wekept the physical parameters fixed at their best-fit values fromTable 2 and only fit for the eclipse depth at 1.25 μm.

The results are shown in Figure 4. Despite the modest qual-ity of the data, the fit is good, with χ2 = 160.7 for 161 dof.We find a depth d2(1.25 μm) of 4.2%±0.4%, corresponding to aJ-band flux ratio d2(1.25 μm)/(1−d2(1.25 μm)) of 4.4% ± 0.4%(see Section 4.2). This value is slightly off from the predictionsbased on our fit to the ULTRACAM data (Figure 5), althoughby less than 2σ .

To derive the eclipse times in Table 1, we used the results ofthe full eclipse fit but fit the model (with the shape parametersheld fixed) to each observation individually.

4. DISCUSSION

The analyses presented in Section 3 show precise determina-tions of the masses and radii of the WDs in the NLTT 11748binary. We have one remaining free parameter, which is the sizeof the radius excess of the CO WD r2, related to the size of itshydrogen envelope. Moreover, we have shown that the eclipsedata are consistent with a linear ephemeris, although there isa systematic shift between the primary and secondary eclipses.Separately, the variation of the secondary eclipse depth withwavelength allows for accurate determination of the temper-ature of the CO WD, which then determines its age through

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The Astrophysical Journal, 780:167 (10pp), 2014 January 10 Kaplan et al.

0.041 0.042 0.043 0.044 0.045R1 (R )

0.10

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

M1

(M)

6.2

6.3

6.4

r2 = 1.00

r2 = 1.02

r2 = 1.04

r2 = 1.06

0.66 0.68 0.70 0.72 0.74 0.76M2 (M )

0.10

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.16

0.18

0.200.22

0.24

Figure 3. Mass and radius constraints as a function of r2, showing r2 = 1.00 (red), 1.02 (blue), 1.04 (green), and 1.06 (purple). In the left panel, we plot M1 vs. R1,while in the right panel we plot M1 vs. M2. In both panels, we plot the best-fit points (filled circles), along with the expected variations according to Section 3.1 (dashedlines). Additionally, we plot contours of constant surface gravity log(g) = 6.2, 6.3, and 6.4 (left) and mass ratio q = 0.16, 0.18, 0.20, 0.22, and 0.24 (right) in black.

(A color version of this figure is available in the online journal.)

Table 2Eclipse Fitting Results

Quantity Value: r2 = 1.00 Value: r2 = 1.02 Value: r2 = 1.04 Value: r2 = 1.06

M1a (M�) . . . 0.136 ± 0.007 0.145 ± 0.007 0.153 ± 0.007 0.162 ± 0.007

R1a (R�) . . . 0.0423 ± 0.0004 0.0426 ± 0.0004 0.0429 ± 0.0004 0.0433 ± 0.0004

K1a,b (km s−1) . . . 273.4 ± 0.5 273.4 ± 0.5 273.4 ± 0.5 273.4 ± 0.5

M2 (M�) . . . 0.707 ± 0.008 0.718 ± 0.008 0.729 ± 0.008 0.740 ± 0.008R2 (R�) . . . 0.0108 ± 0.0001 0.0109 ± 0.0001 0.0110 ± 0.0001 0.0111 ± 0.0001ia (deg) . . . 89.67 ± 0.12 89.67 ± 0.12 89.66 ± 0.12 89.67 ± 0.12a (R�) . . . 1.514 ± 0.009 1.526 ± 0.009 1.538 ± 0.009 1.549 ± 0.009q . . . 0.192 ± 0.008 0.201 ± 0.008 0.210 ± 0.007 0.219 ± 0.007K2 (km s−1) . . . 52.4 ± 2.1 55.0 ± 2.1 57.5 ± 2.0 60.0 ± 2.0R2/R1 . . . 0.2565 ± 0.0006 0.2567 ± 0.0006 0.2568 ± 0.0006 0.2570 ± 0.0006log(g)1 . . . 6.32 ± 0.03 6.34 ± 0.03 6.36 ± 0.03 6.38 ± 0.03log(g)2 . . . 8.22 ± 0.01 8.22 ± 0.01 8.22 ± 0.01 8.22 ± 0.01T1

a,b (K) . . . 8706 ± 136 8705 ± 137 8705 ± 135 8707 ± 136T2

a (K) . . . 7597 ± 119 7594 ± 120 7591 ± 118 7590 ± 119t0a (MBJD) . . . 55772.041585 ± 0.000004 55772.041585 ± 0.000004 55772.041585 ± 0.000004 55772.041585 ± 0.000004PB

a (day) . . . 0.235060485 ± 0.000000003 0.235060485 ± 0.000000003 0.235060485 ± 0.000000003 0.235060485 ± 0.000000003Δta (s) . . . 4.2 ± 0.6 4.2 ± 0.6 4.2 ± 0.6 4.2 ± 0.6χ2/dof . . . 22978.8/22566 22978.8/22566 22978.7/22566 22978.6/22566

Notes.a Directly fit in the MCMC. All other parameters are inferred.b Used a prior distribution based on spectroscopic observations. All other prior distributions were flat.

well-studied CO WD cooling curves. Below, we discuss addi-tional constraints on the masses, radii, and ages of the com-ponents determined by consideration of the eclipse times, sec-ondary temperature, and distance (determined by astrometry).

4.1. Ephemeris, Rømer Delay, and Mass Ratio Constraints

Using the measured eclipse times from Table 1, along withthose reported by Steinfadt et al. (2010), we computed a linearephemeris for NLTT 11748 with a constant frequency fB =1/PB . The residuals (Figure 6) are consistent with being flat andwith the results from the full eclipse fitting (Table 2), showing noindication of orbital changes. However, we do find a systematic

offset between the times of the primary and secondary eclipses,as predicted in Kaplan (2010). The secondary eclipses arriveearlier on average, by Δt = 4.1±0.5 s (after correcting for this,the rms residual for the new data is 1.7 s and the overall χ2 forthe ephemeris data is 38.9 with 29 dof), which is consistent withΔt = 4.2 ± 0.6 s inferred from the full eclipse fitting (Table 2).The sign is correct for a delay caused by the light-travel delayacross the system (the Rømer delay) when the more massiveobject is smaller, the magnitude of which is (Kaplan 2010)

ΔtLT = PBK1

πc(1 − q) . (4)

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0.47 0.48 0.49 0.50 0.51 0.52 0.53Orbital Phase (cycles)

0.90

0.95

1.00

1.05

1.10

Rel

ativ

eF

lux

2010-11-21

2010-12-15

Figure 4. Secondary eclipses of NLTT 11748 observed with Gemini/NIRI.The two observations are the circles/squares, as labeled. The solid curve is thebest-fit model, with the points representing the model integrated over the 20 sexposures for each set of observations.

(A color version of this figure is available in the online journal.)

2000 4000 6000 8000 10000 12000 14000 16000Wavelength (A)

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Flux

Rat

io

u g r i J

Data

Model

Blackbody

Figure 5. Secondary-to-primary flux ratio as a function of wavelength for thesecondary eclipses of NLTT 11748. We plot data from 2010 and 2012 together,along with our best-fit model (open symbols). The different bands are labeled.The model is derived from Tremblay et al. (2011) synthetic photometry. Wealso show the corresponding flux ratio determined from blackbodies (which hadbeen used previously), which does not match the u′ data at all.

(A color version of this figure is available in the online journal.)

If we use Δt = 4.2 ± 0.6 s (from the full eclipse fitting inTable 2), then we infer qLT = 0.29±0.10. This is fully consistentwith our fitted values for q (Table 2; q = 0.192 ± 0.008 forr2 = 1.00). However, it is also possible that some time delay iscaused by a finite eccentricity of the orbit (Kaplan 2010; Winn2011), with

Δte = 2PBe

πcos ω, (5)

where e is the eccentricity and ω is the argument of periastron.14

However, the Rømer delay must be present in the eclipsetiming with a magnitude (4.76 ± 0.05)r2.6

2 s, based on ourmass determination. Therefore, instead of using the Rømer

14 Note that the expression from Kaplan (2010) is missing a factor of two, aspointed out by Barlow et al. (2012).

400 500 600 700 800 900 1000 1100 1200Eclipse Time (MJD-55,000)

−8

−6

−4

−2

0

2

4

6

8

Res

idua

lEcl

ipse

Phas

e(s

)

Mean Primary Eclipse

Mean Secondary Eclipse

Figure 6. Residual eclipse phase vs. eclipse time of NLTT 11748, showing onlythe ULTRACAM measurements, although the measurements of Steinfadt et al.(2010) were included in the ephemeris calculations. The primary eclipses arethe blue squares, while the secondary eclipses are the red circles. The data havebeen fit with a constant-frequency ephemeris. We find an offset between themean time of the primary eclipse compared with that of the secondary eclipseof Δt = 4.1 ± 0.5 s.

(A color version of this figure is available in the online journal.)

delay to constrain the masses, we can use it to constrain theeccentricity. Doing this gives e cos ω = (−4 ± 5) × 10−5

(consistent with a circular orbit). This would then be one ofthe strongest constraints on the eccentricity of any systemwithout a pulsar, as long as the value of ω is not particularlyclose to π/2 (or 3π/2). This may be testable with long-termmonitoring of NLTT 11748, as relativistic apsidal precession(Blandford & Teukolsky 1976) should be ω ≈ 2 deg yr−1 (fora nominal e = 10−3). As long as any tidal precession is ona longer timescale, the change in ω could separate the Rømerdelay from that due to a finite eccentricity. For a system aswide as NLTT 11748 tidal effects are likely to be negligible(Fuller & Lai 2013; Burkart et al. 2013). Further relativisticeffects may be harder to disentangle: an orbital period decayPB will be of a magnitude −1 μs yr−1, compared with a periodderivative from the Shklovskii (1970) effect15 of +24 μs yr−1

(current data do not strongly constrain PB because of the shortbaseline, with PB = (−1900 ± 900) μs yr−1). Therefore, anaccurate determination of the distance (whose uncertaintiescurrently dominate the uncertainty in the period derivative) willbe necessary before any relativistic PB can be measured.

In comparison, the radial velocity constraints on the eccentric-ity are considerably weaker: Steinfadt et al. (2010) determinede < 0.06 (3σ ), while Kilic et al. (2010) and Kawka et al. (2010)both assumed circular orbits. We refit all of the available radialvelocity measurements from those three papers. Assuming acircular orbit, we find K1 = 273.3±0.4 km s−1 (like Kilic et al.2010, whose data dominate the fit). We also tried the eccentricorbit as parameterized by Damour & Taylor (1991). The fit isconsistent with a circular orbit, with e cos ω = 0.001 ± 0.004and e sin ω = −0.004 ± 0.004, which limits e < 0.017 (3σ ).The eclipse durations can also weakly constrain the eccentric-ity, with the ratio of the secondary eclipse to the primary eclipse

15 The Shklovskii (1970) PB (also known as secular acceleration) isparticularly large in the case of NLTT 11748 because of its proximity and largespace velocity (Kawka & Vennes 2009). If measured, it can be used to derive ageometric distance constraint (Bell & Bailes 1996).

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The Astrophysical Journal, 780:167 (10pp), 2014 January 10 Kaplan et al.

duration roughly given by 1 + 2e sin ω (for e 1), althoughas noted by Winn (2011) this is typically less useful than theconstraints on e cos ω from eclipse timing. In the future, we canfit for this term directly.

While all data are satisfactorily fit by just a simple linearephemeris, we can also ask if a third body could be presentin the system. Such a body, especially if on an inclined orbit,could significantly speed up the merger of the inner binary andmay alter the evolution of the system (Thompson 2011). Anyputative tertiary would likely be in a more distant circumbinaryorbit, since the interactions necessary to produce the ELMWD would have disrupted closer companions. A tertiary wouldproduce transit timing variations (Holman & Murray 2005;Agol et al. 2005), moving the eclipse times we measure. Afull analysis of transit timing variations, including nonlinearorbital interactions, is beyond the scope of this paper. Instead,we did a limited analysis where we considered the systemto be sufficiently hierarchical such that the inner binary wasunperturbed (consistent with our measurements) and only itscenter of mass moved due to the presence of the tertiary. Wetook the ephemeris residuals and fit a variety of periodic models,determining for each trial period what the maximum amplitudecould be (marginalizing over phase) such that χ2 increased by 1from the linear ephemeris fit (adding additional terms in generaldecreases χ2, but we wanted to see what the maximum possibleamplitude could be). We found that for periods of 1–300 days,the limit on any sinusoidal component was �1 s (consistent withthe rms discussed above) or smaller than the orbit of the innerbinary. Therefore, unless it is highly inclined, no tertiary withsuch a period is possible. As we get to periods that are longerthan 300 days, we no longer have sufficient data to constrain aperiodic signal, but here the constraints from our polynomial fitalso exclude any stellar-mass companion (an amplitude of 1 sat a period of 300 days would require a mass of 0.002 M� if theouter orbit is also edge-on).

4.2. Secondary Temperature and Age Constraints

As with the NIRI data, we can separate the fitting of thesecondary eclipse depth from the rest of the eclipse fittingand derive the eclipse depths as a function of wavelength,d2(λ) (where we also separate the 2010 and 2012 ULTRACAMobservations). Given the eight secondary eclipse depths thatwe measure, we can determine the ratio of the radius of thesecondary to the primary R2/R1 as well as the temperature ofthe secondary, T2, given measurements of T1. For that, weuse the determination by Kilic et al. (2010)—T1 = 8690 ±140 K—largely consistent with the value determined by Kawkaet al. (2010) of 8580 ± 50 K. The secondary eclipse depths arerelated to the wavelength-dependent flux ratios:

f (λ) ≡ F2(λ)

F1(λ)= d2(λ)

1 − d2(λ)= R2

210−m(λ,T2)/2.5

R2110−m(λ,T1)/2.5

, (6)

where m(λ, T ) is the absolute magnitude of a fiducial WD witha temperature T at wavelength λ.

To determine the secondary’s temperature from the eclipsedepths, we first sample T1 from the distribution N (8690, 140)1000 times. Then, for each sample, we interpolate the syntheticphotometry of Tremblay et al. (2011; using the 0.2 M� model)to determine the photometry for the primary, m(λ, T1). We thensolve for the temperature of the secondary and the radius ratio byminimizing the χ2 statistic, comparing our measured f (λ) withthe synthetic values (using the 0.7 M� grid for the secondary;

the results did not change if we used the 0.6 M� or 0.8 M� gridsinstead).

The resulting distribution of T2 and (R2/R1) is independentof any assumptions in the global eclipse fitting. We findT2 = 7643 ± 94 K and R2/R1 = 0.255 ± 0.002. However,the fits had an average χ2 = 12.6 for 5 dof, mostly comingfrom a small mismatch between the inferred eclipse depth at g′measured in 2010 versus 2012. If we increase the uncertaintiesto have a reduced χ2 = 1, then we find T2 = 7643 ± 150 Kand R2/R1 = 0.255 ± 0.003. Note that the radius ratio here isfully consistent with that inferred from the fit to the rest of theeclipse shape (Section 3 and Table 2).

We show the results in Figure 5 (including the results of theNIRI data analysis), along with the results using blackbodiesfor the flux distributions instead of the synthetic photometry (ashad been done by Steinfadt et al. 2010 and others). It is apparentthat the blackbody does not agree well and using the syntheticphotometry is vital.

The constraints using the separate wavelength-dependenteclipse depths ended up being slightly less precise than thevalue from the full eclipse fitting, although the two constraintsare entirely consistent. We therefore choose the values fromTable 2, where we determined a secondary temperature ofT2 = 7600 ± 120 K. Using the thin (thick) hydrogen atmospheremodels for a 0.7 M� CO WD, we find a secondary ageτ2 = 1.70 ± 0.09 Gyr (1.58 ± 0.07 Gyr) by interpolatingthe cooling curves from Tremblay et al. (2011). So, the envelopeuncertainty does not contribute significantly to the uncertaintyin the age of the secondary directly. A bigger contribution isthrough changes in the secondary mass M2.

4.3. Primary Radius and Distance Constraints

Based on the measured J-band photometry (J =15.84 ± 0.08 from Skrutskie et al. 2006)16, along with an esti-mate for the extinction (E(B − V ) = 0.10 and RV = 3.2, fromKilic et al. 2010), we can compare our measured radius withthat inferred from a parallax measurement (π = 5.6 ± 0.9 mas;H. Harris 2011, private communication). We again use theTremblay et al. (2011) synthetic photometry for the bolomet-ric correction at the temperature determined by Kilic et al.(2010) and use AJ = 0.29AV . Based on these data, we in-fer R1,phot = 0.049 ± 0.009 R�. This is fully consistent (within0.5σ ) with our inferred values from the eclipse fitting. We canuse this value along with the radius ratio inferred from theeclipse shape (roughly R2/R1 = 0.2567 ± 0.0006) to deter-mine R2 = 0.013 ± 0.002 R�. From here, we can calculateM2,thin = 0.57±0.08 M� and M2,thick = 0.61±0.11 M�, whichmakes use of evolutionary models that were interpolated to thecorrect effective temperature (Fontaine et al. 2001; Bergeronet al. 2001). These masses are a little lower than our secondarymasses calculated by the eclipse fitting, but differ by less than2σ . Inverting the problem, we infer based on our eclipse fittingfor r2 = 1.02 a distance d = 159 ± 8 pc (π = 6.3 ± 0.3 mas),with the uncertainty dominated by the uncertainty in the pho-tometry.

Our analysis above included the effects of in-binary mi-crolensing. Steinfadt et al. (2010) assumed microlensing wouldmodify the primary eclipse depth, but did not show definitivelythat it was required for a good fit (cf. Muirhead et al. 2013).

16 Kawka & Vennes (2009) incorrectly give the 2MASS J-band magnitude as15.873 ± 0.077, but the online database lists 2MASS J03451680+1748091 ashaving J = 15.837 ± 0.077.

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The Astrophysical Journal, 780:167 (10pp), 2014 January 10 Kaplan et al.

We fit the same data with the same procedure as in Section 3,but did not allow for any decrease in the depth of the primaryeclipse from lensing. The resulting fit was adequate, althoughslightly worse than with lensing. With no lensing amplification,to match the depth of the primary eclipse requires a reducedvalue of R2/R1 or a lower inclination. We can do that by in-creasing R1 or decreasing R2, but that is difficult as the eclipsedurations fix R1/a and R2/a. We end up accomplishing thisby increasing the masses, which widens the orbit (increasing ato go along with the increase in R1 and decreasing R2 throughthe mass-radius relation for the WD). For r2 = 1.00, we findM1 = 0.16 M� and M2 = 0.74 M�, along with R1 = 0.044 R�(log(g)1 = 6.36 ± 0.04). While this combination of param-eters does not give as good a fit to the data as the fit withlensing, the difference is not statistically significant, with anincrease in χ2 of 20 (the reduced χ2 increased from 1.018 to1.019, for a chance of occurrence of about 50%). We wouldneed other independent information (such as a much moreprecise parallax or a more precise time delay) to break thedegeneracies.

5. CONCLUSIONS

Using extremely high-quality photometry combined withimproved modeling, we have determined the masses and radiiof the WDs in the NLTT 11748 binary to better than ±0.01 M�and ±0.0005 R� statistical precision, although uncertainties inthe radius excess limit our final precision. This analysis makesuse of the eclipse depth and shape, including corrections forgravitational lensing, and is consistent with the Rømer delaymeasured independently from eclipse times, Δt = 4.2 ± 0.6 s.This would be the first detection of an observed Rømer delay forground-based eclipse measurements (cf. Bloemen et al. 2012;Barlow et al. 2012), although in all of these systems there is thepossibility that the time delay is instead related to a finite, butsmall, eccentricity.

Our mass measurement for the smallest plausible radiusexcess (r2 = 1.02), M1 = 0.137 ± 0.007 M�, is significantlylower than that inferred by Kilic et al. (2010) on the basis ofthe Panei et al. (2007) evolutionary models or that inferred byAlthaus et al. (2013) from newer models. Even for the highestvalue of r2 that we considered (1.06), we still find a primary massof 0.157 ± 0.008 M�, significantly below the 0.17–0.18 M�range discussed by Kilic et al. (2010) and Althaus et al. (2013).This may call for a revision of those models to take into accountthe improved observational constraints or it may indicate that aneven higher value of r2 is more realistic. In any case, our surfacegravity is lower than that from Kilic et al. (2010), which wasused by Althaus et al. (2013), while it is consistent to within1σ with the gravity measured by Kawka et al. (2010). Withour log(g) determination, the mass is closer to the predictionfrom Althaus et al. (2013, who find M = 0.174 M� forlog(g) = 6.40 and log10 Teff = 3.93, compared with 0.183 M�for the higher log(g)), although again high values of r2 arerequired. However, our cooling age for the secondary is a factorof two to three smaller than the cooling age of >4 Gyr for theprimary predicted by the Althaus et al. (2013) models for thelower mass. This might be a reflection of the non-monotonicevolution experienced by some ELM WDs, even though NLTT11748 seems to have a low enough mass that it would cool ina simpler manner. Further constraints on the mass ratio fromeclipse timing or direct detection of the secondary’s spectrum(the inferred values of K2 in Table 2 vary significantly) couldhelp resolve this question.

We thank the anonymous referee for helpful comments. Thiswork was supported by the National Science Foundation un-der grants PHY 11-25915 and AST 11-09174. T.R.M. wassupported under a grant from the UK’s Science and Technol-ogy Facilities Council (STFC), ST/F002599/1. V.S.D., S.P.L.,and ULTRACAM were supported by the STFC. A.N.W. wassupported by the University of Wisconsin, Milwaukee Officeof Undergraduate Research. We thank the staff of Gemini forassisting in planning and executing these demanding observa-tions. We thank D. Foreman-Mackey for help using emcee andL. Althaus for supplying evolutionary models. Based on ob-servations obtained at the Gemini Observatory, which is op-erated by the Association of Universities for Research in As-tronomy, Inc., under a cooperative agreement with the NSF onbehalf of the Gemini partnership: the National Science Founda-tion (United States); the National Research Council (Canada);CONICYT (Chile); the Australian Research Council (Aus-tralia); Ministerio da Ciencia, Tecnologia, e Inovacao (Brazil);and Ministerio de Ciencia, Tecnologıa, e Innovacion Productiva(Argentina).

Facilities: Gemini:Gillett (NIRI), NTT (ULTRACAM),ING:Herschel (ULTRACAM)

REFERENCES

Agol, E. 2002, ApJ, 579, 430Agol, E., Steffen, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567Althaus, L. G., Miller Bertolami, M. M., & Corsico, A. H. 2013, A&A,

557, A19Antoniadis, J., van Kerkwijk, M. H., Koester, D., et al. 2012, MNRAS,

423, 3316Barlow, B. N., Wade, R. A., & Liss, S. E. 2012, ApJ, 753, 101Bassa, C. G., van Kerkwijk, M. H., Koester, D., & Verbunt, F. 2006, A&A,

456, 295Bell, J. F., & Bailes, M. 1996, ApJL, 456, L33Bergeron, P., Leggett, S. K., & Ruiz, M. T. 2001, ApJS, 133, 413Bergeron, P., Saffer, R. A., & Liebert, J. 1992, ApJ, 394, 228Blandford, R., & Teukolsky, S. A. 1976, ApJ, 205, 580Bloemen, S., Marsh, T. R., Degroote, P., et al. 2012, MNRAS, 422, 2600Brown, W. R., Kilic, M., Allende Prieto, C., Gianninas, A., & Kenyon, S. J.

2013, ApJ, 769, 66Brown, W. R., Kilic, M., Hermes, J. J., et al. 2011, ApJL, 737, L23Burkart, J., Quataert, E., Arras, P., & Weinberg, N. N. 2013, MNRAS, 433, 332Carter, J. A., & Winn, J. N. 2009, ApJ, 704, 51Claret, A. 2000, A&A, 363, 1081Damour, T., & Taylor, J. H. 1991, ApJ, 366, 501D’Antona, F., Ventura, P., Burderi, L., & Teodorescu, A. 2006, ApJ,

653, 1429Deloye, C. J., Bildsten, L., & Nelemans, G. 2005, ApJ, 624, 934Dhillon, V. S., Marsh, T. R., Stevenson, M. J., et al. 2007, MNRAS, 378, 825Fontaine, G., Brassard, P., & Bergeron, P. 2001, PASP, 113, 409Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP,

125, 306Fuller, J., & Lai, D. 2013, MNRAS, 430, 274Gianninas, A., Strickland, B. D., Kilic, M., & Bergeron, P. 2013, ApJ, 766, 3Goodman, J., & Weare, J. 2010, Comm. App. Math. Comp. Sci., 5, 65Hobbs, G. B., Edwards, R. T., & Manchester, R. N. 2006, MNRAS, 369, 655Hodapp, K. W., Jensen, J. B., Irwin, E. M., et al. 2003, PASP, 115, 1388Holman, M. J., & Murray, N. W. 2005, Sci, 307, 1288Iben, I., Jr., & Tutukov, A. V. 1984, ApJS, 54, 335Kaplan, D. L. 2010, ApJL, 717, L108Kaplan, D. L., Bhalerao, V. B., van Kerkwijk, M. H., et al. 2013, ApJ,

765, 158Kaplan, D. L., Bildsten, L., & Steinfadt, J. D. R. 2012, ApJ, 758, 64Kawka, A., & Vennes, S. 2009, A&A, 506, L25Kawka, A., Vennes, S., & Vaccaro, T. R. 2010, A&A, 516, L7Kilic, M., Allende Prieto, C., Brown, W. R., et al. 2010, ApJL, 721, L158Kilic, M., Brown, W. R., Allende Prieto, C., et al. 2012, ApJ, 751, 141Lorimer, D. R., Lyne, A. G., Festin, L., & Nicastro, L. 1995, Natur, 376, 393Maeder, A. 1973, A&A, 26, 215Mandel, K., & Agol, E. 2002, ApJL, 580, L171Marsh, T. R. 2001, MNRAS, 324, 547

9

The Astrophysical Journal, 780:167 (10pp), 2014 January 10 Kaplan et al.

Marsh, T. R., Dhillon, V. S., & Duck, S. R. 1995, MNRAS, 275, 828Marsh, T. R., Nelemans, G., & Steeghs, D. 2004, MNRAS, 350, 113Moffat, A. F. J. 1969, A&A, 3, 455Muirhead, P. S., Vanderburg, A., Shporer, A., et al. 2013, ApJ, 767, 111Panei, J. A., Althaus, L. G., Chen, X., & Han, Z. 2007, MNRAS, 382, 779Parsons, S. G., Marsh, T. R., Gansicke, B. T., Drake, A. J., & Koester, D.

2011, ApJL, 735, L30Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4Romero, A. D., Corsico, A. H., Althaus, L. G., et al. 2012, MNRAS, 420, 1462Serenelli, A. M., Althaus, L. G., Rohrmann, R. D., & Benvenuto, O. G.

2002, MNRAS, 337, 1091Shklovskii, I. S. 1970, SvA, 13, 562Shporer, A., Kaplan, D. L., Steinfadt, J. D. R., et al. 2010, ApJL, 725, L200

Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163Steinfadt, J. D. R., Kaplan, D. L., Shporer, A., Bildsten, L., & Howell, S. B.

2010, ApJL, 716, L146Tauris, T. M., Langer, N., & Kramer, M. 2012, MNRAS, 425, 1601Thompson, T. A. 2011, ApJ, 741, 82Tremblay, P.-E., & Bergeron, P. 2008, ApJ, 672, 1144Tremblay, P.-E., Bergeron, P., & Gianninas, A. 2011, ApJ, 730, 128van Kerkwijk, M. H., Bassa, C. G., Jacoby, B. A., & Jonker, P. G. 2005, in ASP

Conf. Ser. 328, Binary Radio Pulsars, ed. F. A. Rasio & I. H. Stairs (SanFrancisco, CA: ASP), 357

Vennes, S., Thorstensen, J. R., Kawka, A., et al. 2011, ApJL, 737, L16Webbink, R. F. 1984, ApJ, 277, 355Winn, J. N. 2011, in Exoplanet Transits and Occultations, ed. S. Piper (Tucson,

AZ: Univ. Arizona Press), 55

10